On primitive Pythagorean triples of special forms

TL;DR

This paper investigates specific families of primitive Pythagorean triples with particular forms, providing methods to generate infinitely many and analyzing their density among all primitive triples of the same parity.

Contribution

It introduces a systematic way to generate infinite primitive triples of special forms and studies their asymptotic density within all primitive triples.

Findings

Infinite primitive triples can be generated for each fixed g and f.

Asymptotic density of these triples is characterized for each g.

The methods extend understanding of the structure of primitive Pythagorean triples.

Abstract

We explore primitive Pythagorean triples of special forms $(a,b,b+g)$ and $(a,a+f,c)$, with $g,f\in\mathbb{Z}^+$. For each $g$ and $f$, we provide a method to generate infinitely many such primitive triples. Lastly, for each $g$, we describe the asymptotic density of primitive $(a,b,b+g)$ triples within all primitive triples of the same parity.